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Title:
Transducer dynamics
Creator:
Dolzhenko, Egor
Place of Publication:
[Tampa, Fla]
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University of South Florida
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## Subjects

Subjects / Keywords:
Sequences of words
Finite state automata with output
Entropy
Picture languages
Local languages
Dissertations, Academic -- Mathematics -- Masters -- USF ( lcsh )
Genre:
non-fiction ( marcgt )

## Notes

Abstract:
ABSTRACT: Transducers are finite state automata with an output. In this thesis, we attempt to classify sequences that can be constructed by iteratively applying a transducer to a given word. We begin exploring this problem by considering sequences of words that can be produced by iterative application of a transducer to a given input word, i.e., identifying sequences of words of the form w, Ï„(w), Ï„²(w), . . . We call such sequences transducer recognizable. Also we introduce the notion of "recognition of a sequence in context", which captures the possibility of concatenating prefix and suffix words to each word in the sequence, so a given sequence of words becomes transducer recognizable. It turns out that all finite and periodic sequences of words of equal length are transducer recognizable. We also show how to construct a deterministic transducer with the least number of states recognizing a given sequence. To each transducer Ï„ we associate a two-dimensional language L²(Ï„) consisting of blocks of symbols in the following way. The first row, w, of each block is in the input language of Ï„, the second row is a word that Ï„ outputs on input w. Inductively, every subsequent row is a word outputted by the transducer when its preceding row is read as an input. We show a relationship of the entropy values of these two-dimensional languages to the entropy values of the one-dimensional languages that appear as input languages for finite state transducers.
Thesis:
Thesis (M.A.)--University of South Florida, 2008.
Bibliography:
Includes bibliographical references.
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Title from PDF of title page.
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Document formatted into pages; contains 31 pages.
Statement of Responsibility:
by Egor Dolzhenko.

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E14-SFE0002380 ( USFLDC DOI )
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USF Electronic Theses and Dissertations

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ABSTRACT: Transducers are finite state automata with an output. In this thesis, we attempt to classify sequences that can be constructed by iteratively applying a transducer to a given word. We begin exploring this problem by considering sequences of words that can be produced by iterative application of a transducer to a given input word, i.e., identifying sequences of words of the form w, (w), (w), . We call such sequences transducer recognizable. Also we introduce the notion of "recognition of a sequence in context", which captures the possibility of concatenating prefix and suffix words to each word in the sequence, so a given sequence of words becomes transducer recognizable. It turns out that all finite and periodic sequences of words of equal length are transducer recognizable. We also show how to construct a deterministic transducer with the least number of states recognizing a given sequence. To each transducer we associate a two-dimensional language L() consisting of blocks of symbols in the following way. The first row, w, of each block is in the input language of , the second row is a word that outputs on input w. Inductively, every subsequent row is a word outputted by the transducer when its preceding row is read as an input. We show a relationship of the entropy values of these two-dimensional languages to the entropy values of the one-dimensional languages that appear as input languages for finite state transducers.
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Finite state automata with output
Entropy
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Local languages
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Listofgures1Sequenceofperiod8andtransducerrecognizingit.................102TransducerusedinProposition3:3.........................133TransducerthatrecognizessetMwithaip.....................144TransducersuchthatLislocal,butL2isnot...............255Transducergeneratingalocalpicturelanguage....................266Transducergeneratingnon-locallanguage.......................29 ii

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0 0 0 1 0 1 q0 q1 ToeachtransitionofthistransducerwecanassociateWangprototilesasfollows. q0 q0 0 0 0 q0 0 q0 q1 q1 1 1 Whereeachstateandeachsymbolofthealphabetrepresentsadistinctcolor.Thesetilescanbeassembledintotherowdepictedbelow. 0 1 q1 q0 0 0 q0 q0 0 1 q0 q1 0 0 q0 q0 Noticethatthebottomedgeofthisrowrepresentsinputword0100andtopedgerepresents0010,thatis,0010istheoutputofthetransducerontheinput0100.Sim-ilarlywecanconstructanotherrowoffourtiles,withbottomedgerepresentingword0010andthetopedgeword0001bystackingthisrowontopoftherst.Continuinginthiswaywecanconstructablockofarbitraryheight.Theaboveexampleillustratesthemaingoalofthiswork,theclassicationofpatternsthatcanbegeneratedbythedescribedprocess.Webeginexploringthisproblembyconsideringsequencesofwordsthatcanbeproducedbyiterativeappli-cationofatransducertoagiveninputword,i.e.,identifyingsequencesofwordsoftheformw;w;2w;:::.Wecallsuchsequencestransducerrecognizable.Alsoweintroducenotionofrecognitionofasequenceincontext",whichgivesrisetoapossibilityofconcatenatingprexandsuxtoeachwordinthesequence,soagivensequenceofwordsbecomestransducerrecognizable.Itturnsoutthatallniteandperiodicsequencesofwordsofequallengtharetransducerrecognizable.Additionallywebrieyexploreotherwaystoiterativelyapplyatransducertoawordinordertogetasequence.Thenextquestionwhichweconsideristhefollowing:Givenasequenceofwordss,howcanoneconstructadeterministictransducerwiththeleastnumberofstates 2

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Weassumethatanytransducercontainsatmostonestateqjunk. Denition1.5. Foratransducer=A;Q;;;q0;Fandw2Aifacceptswandq0;w=u,thenwewritew=u.LetRngdenotethesetofallwordsusuchthatthereisw,suchthatw=u. Denition1.6. Anondeterministictransducerisave-tuple=A;Q;;q0;FwhereAisanitealphabet,Qisanitesetofstates,q0isaninitialstateq02Q,FisasetofnalstatesFQandisatransitionrelationQAAQ.Notethatalloftheabovenotionsaredenedsimilarlyfornondeterministictransducers,however,inthedenitionofthenondeterministictransducer,isnotafunction.Forinstance,givenwinL,wnowdenesaset,sincemayhavemorethanoneoutputonw. 5

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2RecognitionincontextIn[2]theauthorsdescribeawaytouseWangtilestosimulateatransduceronaninputwordw.Briey,eachtilerepresentsaspecictransition,withcolorsoftheleftandrightedgesencodingsourceandtargetstates,andtopandbottomcolorsencodinginputandoutputsymbols.Thereisarowofjwjtileswiththebottomedgeencodingwordw,suchthatleftmostverticaledgeencodesq0,initialstateof,therightmostverticaledgeofthisrowisoneoftheterminalstatesof,andalloftheadjacentverticaledgesencodethesamestate.Thisimpliesthattopedgeofthisrowencodesw,theoutputofoninputw.Notethatwhenisdeterministic,therowwiththepropertiesaboveisunique.Incasethatwordwisacceptedby,thereisanotherrowwiththepropertiesmentionedabove,suchthatitstopedgeiswandthebottomedgeisw.Continuinginthiswayandplacingtheserowsontopofoneanothertwodimensionalblockisobtained,correspondingtothesequencew,w,2w;:::.Themaingoalofthissectionistoformalizetheabovediscussionthroughthenotionsofrecognitionandrecognitionincontextandtodiscusswhatsequencesofwhatperiodscanbeobtainedthroughsuchprocess.Furthermore,lastsectiondiscussessetsofsequencessuchthateachsequenceinthesetisoftheforms1,s1,2s1;:::,wheres1denotesitsrstelement.Forexamplethismaybethecasewhenallsequenceshavesimilarstructureanddieronlyinthelengthoftheirwords.2.1RecognitionThissectiondealsonlywithdeterministictransducers. Denition2.1. Ifs=s1;s2;:::;sk,isanitesequenceofwords,thenwewrite#s=kandifsisinnite,then#s=1. Denition2.2. Asequences=s1;s2;:::iscalledperiodicifthereexistsp2Nsuchthatforallsi=sp+i,i2N.Theleastsuchpiscalledtheperiodofs. 6

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Denition2.3. Lets=s1;s2;s3;:::beasequenceofwordsoveralphabetAsuchthatjsij=jsjjforalli;j.Ifthereexistsadeterministictransducersuchthatsi+1=sifor1i<#s,thensissaidtobetransducerrecognizableandissaidtorecognizes. Denition2.4. Atransducerrecognizespreciselyasequencesifrecognizessandforanyothersequencet,suchthatrecognizest,thereisanaturalnumbern,suchthatns1=t1.Heres1andt1denoterstelementsofsequencessandtrespectively. Proposition2.1. Lets=s1;s2;:::beasequenceofwordsoverAwithjsij=jsjj=kforall1i;j<#s.Thenthissequenceistransducerrecognizableifandonlyiffollowingholds:Forallr=1;:::;k;if8t=1;:::;r,sit=sjtthensi+1r=sj+1rforalli;j. Proof. Incaseholds,considerthetransducernotethatdenotesanemptyword=A;Q;;;;Fthatwouldrecognizes,whereQ,thesetofstates,isdenedbyQ=fg[fsi:::sitjsi2sand1tkg:sincekisxed,thissetisnite.F=fsijsiisamemberofasequencesgForeverysi2ssuchthat1i<#sdenesi:::sit)]TJ/F15 11.955 Tf 11.956 0 Td[(1;sit=si:::sitandsi:::sit)]TJ/F15 11.955 Tf 11.955 0 Td[(1;sit=si+1t: 7

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Also,add;si=siand;si=si+1.Thentheoutputsymbolisuniquelydeterminedbysi:::sitdueto.Henceisdeterministic.FinallyaddqjunktoQandletallthemissingtransitionsleadtoit.Notethatq0;w2F,i.e.,thistransduceracceptswordwifandonlyifw2sandthat;si=si+1byconstruction.Thusrecognizess.Conversely,ifforsomesi;sj2swith1i;j<#sandforsomelhavesi:::sil=sj:::sjlbutsi+1l6=sj+1landthereisdeterministictrans-ducer=A;Q;;;q0;Fthatrecognizessthenletq=q0;si:::sil)]TJ/F15 11.955 Tf 12.895 0 Td[(1=q0;sj:::sjl)]TJ/F15 11.955 Tf 13.554 0 Td[(1.Thisisacontradiction,sincesi+1l=q;sil6=q;sjl=sj+1l. Ifsisatransducerrecognizablesequence,letsdenotethetransducercon-structedbythealgorithmintheproposition2.1thatrecognizess.Notesomeofthepropertiesofs: i. Ifq0;w2F,thenbyconstructionofs,q0;w=wandw2s ii. Ifq0;w0=q0;w00,thenw0=w00,sincebyi,w0=q0;w0=q0;w00=w00. iii. Thetransducersrecognizessprecisely,sincerecognizessandforanyothersequence,w;w;2w:::,thatrecognizes,wmustbeacceptedbyandthen,byi,w2s.SupposejAj=2andletsbeatransducerrecognizableperiodicsequenceoveralphabetAofperiodgreaterthanone.Thentheperiodofsmustbeeven.ToseethissupposeA=fa;bg.Lettbetheleastnaturalnumbersuchthatthereisiandj,sit6=sjt.Thenitfollowsthats1ts2ts3t:::=ababab:::ors1ts2ts3t:::=bababa:::.Thisisso,sinceduetodeterminismof,itmustbetruethatq0;si:::sit)]TJ/F15 11.955 Tf 12.076 0 Td[(1=q0;sj:::sjt)]TJ/F15 11.955 Tf 12.075 0 Td[(1=:q0foralli;j.First,q0;a=aandq0;b=bcannothappenbythechoiceoft,sinceaboveimpliesthatsit=s1tforalli.Second,q0;a=aandq0;b=acan'thappeneither,sinceinthiscasesit=aforalliors1t=bandsit=a,i>1inwhichcasethesequenceisnotevenperiodic.Thusoneofthetwocasesmentionedabovemustbetrue,whichimpliesthatperiodofthesequencesmustbeeven. 8

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Proposition2.2. Foreachnaturalnumbern,thereexistsatransducerrecognizablesequencefsig10overalphabetA=fa0;a1;:::;akgofperiodjAjn. Proof. byinductiononnForn=1,letsi=aimodjAj.Sincesi=sjimpliesthataimodjAj=ajmodjAjandijmodjAjwehavethati+1j+1modjAjandsi+1=sj+1.Thusthissequencesatisestheconditionsoftheorem2.1,andthusitistransducerrecognizableandclearlyofperiodjAj.Assumethatthepremiseholdsforn=t)]TJ/F15 11.955 Tf 11.955 0 Td[(1,i.e.thatthereexistsasequencefs0ig10ofperiodjAjt)]TJ/F19 7.97 Tf 6.587 0 Td[(1.Forn=jAjtletsi=s0iammodjAjsuchthati=mjAjt)]TJ/F19 7.97 Tf 6.586 0 Td[(1+rwhere0r
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q0 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 1 0 1 0 0 0 1 1 1 1 0 0 0 0 1 1 1 0 1 0 0 1 1 0 0 0 1 1 011 101 001 110 010 100 000 111 Figure1:Sequenceofperiod8andtransducerrecognizingit denenewsequencefsigk)]TJ/F19 7.97 Tf 6.586 0 Td[(10bypi=bi,i=0:::k)]TJ/F15 11.955 Tf 12.619 0 Td[(1.Thenfwigisrecognizableincontextwithprexfpig.Toseethisnotethatifpiwi=pjwjthenpi=pj,hencei=j.Inotherwordssequencefpiwigk1sucesthepremiseoftheProposition2.1. Corollary2.2. EverysequenceconsistingofwordsofequallengthofperiodjAjnisrecognizableincontext. Proof. LetfwigbethesequenceofperiodjAjn.Theorem2.2yieldstransducerrec-ognizablesequencesequencefpigofperiodjAjn.Hence,similarlytothepreviouscorollary,sequencefwgisrecognizableincontextwithprexfpig. Denition2.6. LetD=fs1;s2;s3;:::gbeasetofsequencesofwordsofequallengthoveralphabetA.Thenifthereexistsadeterministictransducer,suchthatrecognizessiincontext,foralli=1;:::;jDj,withthesameprexandsuxforeverysetsi,thenwesaythatrecognizessetDandthatsetDisrecognizableincontext.Thiswaywerequireofonetransducertorecognizeasetofsequenceswiththesameprexandsux. Example2.1. SetM=ff0n)]TJ/F22 7.97 Tf 6.586 0 Td[(i10i)]TJ/F19 7.97 Tf 6.586 0 Td[(1gni=1jn=2;3;:::gisnotrecognizableincontext. 10

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Proof. SupposethatthereexistssuchthatrecognizesMincontextwithprexpandsuxd.Considertherstelementofthesequencef0n)]TJ/F22 7.97 Tf 6.586 0 Td[(i10i)]TJ/F19 7.97 Tf 6.587 0 Td[(1gni=12M,suchthatn)]TJ/F15 11.955 Tf 11.254 0 Td[(1isgreaterthantwicethenumberofstatesof.Thenitfollowsthatonthesubword0n)]TJ/F19 7.97 Tf 6.587 0 Td[(11ofp10n)]TJ/F19 7.97 Tf 6.586 0 Td[(11d1transducergoesthroughstatesqt1;qt2;qt3;:::;qtn+1andyields0n)]TJ/F19 7.97 Tf 6.587 0 Td[(210,i.e., qt1;0=;qt2,qt2;0=;qt3,:::qtn)]TJ/F20 5.978 Tf 5.756 0 Td[(1;0=1;qtn;qtn;1=0;qtn+1:Bythepigeonholeprinciple,therearetm;tlsuchthatqtm=qtlandtm
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Denition2.7. Lets=s1;s2;s3;:::beasequenceofwordsoveralphabetAsuchthatforalli;jjsij=jsjj.Ifthereexistsadeterministictransducerandsequencep=p1;p2;p3;:::suchthat8i;jjpij=jpjj,andsequenced=d1;d2;d3;:::with8i;jjdij=jdjjand#d=#p=#ssuchthatpi+1si+1di+1=pisidiRRfori=1;2;3;:::;#s)]TJ/F15 11.955 Tf 11.899 0 Td[(1.Thenissaidtorecognizeswithaipincontextandsisrecognizablewithaipincontext.Andhencewecancorrespondinglyadjustdenitionforrecognitionofaset. Denition2.8. LetD=fs1;s2;s3;:::gbeasetofsequencesofwordsoveralphabetA.Thenifthereexistsadeterministictransducersuchthatrecognizessiwithaipincontextforalli=1:::jDj,withthesameprexandsuxforeverysetsi,thenwesaythatrecognizessetDwithaipincontextandthatsetDisrecognizablewithaipincontext. Proposition2.3. Alltransducerrecognizablesequencesarerecognizablewithaipincontext. Proof. Lets=s1;s2;s3:::beatransducerrecognizablesequenceofwordsoverA.Let=A;Q;;;q0;Fbethetransducerthatrecognizess.Usingletsdene0=A;Q0;0;0;q00;f0asitisdoneinFigure2.2.Letprexpandsuxdbedenedbypi=0anddi=1foralli.Thisway0willbeabletodistinguishbetweenwandwRforeachw2s.Since00pisidiRR=00si1RR=00sRi0R=0sRi0R=0si1=0si+11,itfollowsthatsisrecognizablewithaipincontext. Example2.2. M=ff0n)]TJ/F22 7.97 Tf 6.586 0 Td[(i10i)]TJ/F19 7.97 Tf 6.586 0 Td[(1gni=1jn=2;3;:::gisrecognizablewithaipincon-text. Proof. Lets2M.Thuss=f0t)]TJ/F22 7.97 Tf 6.586 0 Td[(i10i)]TJ/F19 7.97 Tf 6.586 0 Td[(1gti=1forsomet.Thensi=0t)]TJ/F22 7.97 Tf 6.587 0 Td[(i10i)]TJ/F19 7.97 Tf 6.586 0 Td[(1istheithwordinthesequences.Deneprexpandsuxdbypi=1anddi=0foreachi.LettransducerbeasitisdenedintheFigure2.2.Then 12

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q00 ::: q01 q02 q0k qt1 q0 qt2 q0k+2 q0k+1 Figure2:TransducerusedinProposition3:3 q5 q6 q0 q3 1 q4 q1 1 1 0 1 0 0 0 0 1 1 1 0 q2 0 0 0 0 q7 1 1 1 0 0 Figure3:TransducerthatrecognizessetMwithaip wRiR=0t)]TJ/F22 7.97 Tf 6.587 0 Td[(i10i)]TJ/F19 7.97 Tf 6.587 0 Td[(10RR=0i)]TJ/F19 7.97 Tf 6.587 0 Td[(110t)]TJ/F22 7.97 Tf 6.586 0 Td[(i1R=0i10t)]TJ/F22 7.97 Tf 6.586 0 Td[(i)]TJ/F19 7.97 Tf 6.587 0 Td[(11R=0t)]TJ/F22 7.97 Tf 6.586 0 Td[(i)]TJ/F19 7.97 Tf 6.587 0 Td[(110i1=10t)]TJ/F22 7.97 Tf 6.587 0 Td[(i)]TJ/F19 7.97 Tf 6.586 0 Td[(110i0=1si+10.ThussetMisrecognizablewithaipincontext. 13

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3MinimizationAMealymachineisavetuple=A;Q;;;q0whoseelementshavethesamedenitionastherstveelementsofthedeterministictransducer.Hencede-terministictransduceraresomewhatmoregeneralthanMealymachines,whichhavebeenstudiedwell.Howeveraimofthefollowingsectionistoshowthatwhenitcomestominimizationthesamealgorithmcanbeapplied.Thesecondsectiondealswithndinginsomesensesmallestdeterministictransducer,ifoneexists,thatcanrecognizegivensequence.3.1MinimizationofaDeterministicTransducerThedenitionofthedeterministictransducerdescribedaboveismoregeneralthanthatoftheMealymachine,however,onlydierenceisthepresenceofthenalstatesindeterministictransducer,whichcanbeconrmedin[4].Thusitseemsplausibletousealgorithm,similartothealgorithmfortheminimizationofaMealymachine,thatwouldtakeintoaccountthesetofthenalstates.Hencethecontentofthissectionis,althoughmodiedformoregeneralcase,takenfrom[4]. Denition3.1. Astateaoftransducer=A;Q;;;q0;Fisaccessibleifthereissomeinputwordw2Asuchthatq0;w=q.Transducerisconnectedifeverystateisaccessible.Forsimplicityassumethatalltransitionsgoingtothestateqjunkhaveinputsymbolequaltotheoutputsymbol,i.e.q;a=a;qjunk,whereqjunkisthestateforwhich8w2Aqjunk;w=2F.Also,withoutlossofgenerality,itisassumedthatqjunkistheonlystatewiththispropertyandthateverytransducerisconnected. Denition3.2. Foratransducerandw2A,letw=uifacceptswandoutputsu. Denition3.3. Twotransducers1=A1;Q1;1;1;q01;F1 14

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and2=A2;Q2;2;2;q02;F2areequivalentifandonlyif 1. A1=A2,and 2. L1=L2andforallw2L1w=2w.Asarelation,equivalenceestablishesanequivalencerelationonasetofalldeterministictransducers. Denition3.4. Twostatesqaandqboftransducer=A;Q;;;q0;Fareequiv-alentifandonlyifa=A;Q;;;qa;Fisequivalenttob=A;Q;;;qb;F. Proposition3.1. Twostatesqaandqbofatransducer=A;Q;;;q0;Fareequivalentifandonlyif 1. 8s2Aqa;s=qb;s,and 2. 8s2Aqa;sisequivalenttoqb;s. Proof. Ifbothoftheconditionshold,thenletw2Aandw=snwheres2A,thent=qa;s=qb;sandqa;sn=qb;sn=k.Thusqaw=qbw=tk.Conversely,letqabeequivalenttoqb,thenletw=sn,wheres2Aandn2A.Sinceasn=bsnitfollowsthatqa;sn=qb;snhenceqa;sisequivalenttoqb;s.Alsoqa;s=qb;ssinceifqa;s=2qjunkthereexistsw0suchthatqa;sacceptsw0andhenceqaacceptssw0,implyingthatqasw0=qbsw0,thusqa;s=qb;s.Ifqa;s=qjunkandqb;s=qjunkthenqa;s=qb;s=sbypreviousassumptionthattransitionsleadingtoqjunkproduceoutputequaltoinput. Denition3.5. Atransducerisreducedifitcontainsnopairofequivalentstates. 15

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Denition3.6. Statesqaandqbofatransducer=A;Q;;;q0;Farek-distinguishableifthereexistsawordw2Ajwjk,suchthatqaw6=qbw:Thenwiscalledadistinguishingword. Denition3.7. Iftwostatesqaandqbarenotkdistinguishable,thentheyarek-equivalent. Proposition3.2. Twostatesqaandqbofatransducerarek-equivalentifandonlyif 1. Theyare1-equivalent. 2. Foreachs2Aqa;sandqb;sarek)]TJ/F15 11.955 Tf 11.955 0 Td[(1equivalent. Proof. Essentiallythesameasinpreviousproposition. Iftwostatesofatransducerarek-equivalentforallkthentheyareequivalent.Thisrelationdenesapartitionofthesetofthestatesofatransducer,whichisusedforconstructionofanew,reducedtransducer.Thealgorithmgivenin[4]couldbeusedinspitethefactthatitwaswrittenforMealymachines,aslongasappropriatedenitionsareused.3.2MinimalTransducerthatRecognizesSequenceofWordsIntheprevioussectionwetriedtominimizethenumberofstatesinthedeter-ministictransducer.Forinstancegivenatransducerrecognizablesequenceofwordssthealgorithmin[4]canbeappliedtostondanequivalenttransducer,butwiththeminimalnumberofstates.Ontheotherhandif,foragivensequences,weneedtondatransducerwiththeminimalnumberofstatesthatwouldrecognizes,theabovealgorithmwouldnotwork,astheremaybeothertransducers,notequivalenttosthatalsorecognizes.Probablythemostbasicalgorithmforndingtheminimaltransducerthatwouldrecognizeagivensequencescouldproceedasfollows: 1. ForagivensequencesconstructsusingalgorithmoutlinedinProposition2.1. 16

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2. Sincethereareonlynitelymanytransducersovernitealphabetwithxednumberofstates,enumeratealltransducersthathavefewerstatesthanthetransducerconstructedinstep1. 3. SincescannotbeaperiodictobetransducerrecognizableandthereareonlyjAjkdierentwordsoflengthkoveralphabetA,atransducerwiththeminimalnumberofstatesrecognizingscanbefoundinnitelymanysteps.Intheprevioussection,anequivalencerelationonthesetofstateswasusedtodeterminethestatesthatareequivalentandanewtransducer-obtainedthroughequivalenceclassesofthatrelation-wasequivalenttotheoriginalone.Here,aslightlydierentapproachmustbetaken.Theideaistoexamineanewrelationonthesetofstatesthatwouldindicatethestatesthatinsomesensecanbejoinedtogetherwithoutaectingthetransducer'sabilitytorecognizeagivensequence.Let0denotethetransducerconstructedfrombymakingallofthestatesof,exceptqjunk,nal.Thenletdomdenoteallofthewordsacceptedby0. Denition3.8. Letq1;q22Q,thenstatesq1andq2areinrelationq1q2ifandonlyifforallw2domq1domq2havethatq1;w=q2;w. Proposition3.3. Ifq1q2ands2domq1domq2Athenq1;sq2;s. Proof. Letq1,q2,sbeasdescribedinpremiseand:q1;sq2;s.Thenitfollowsthat9w2domq1;sdomq2;s,suchthatq1;s;w6=q2;s;w.Thensw2domq1domq2andq1;sw=q1;sq1;s;w6=q2;sq2;s;w=q2;sw,whichisacontradiction. Proposition3.4. Letbeadeterministictransducerrecognizingthesequences=s1;s2;:::,andletEfu;vju;v2Qanduvgbesuchthat: EdenesanequivalencerelationonsetofstatesQ 8u;v2E,ifa2domudomvA,thenu;a;v;a2E.Thenthereexists0,withsetofstatesequaltothesetoftheequivalenceclassesproducedbyE,suchthat0recognizess. 17

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Proof. LetQ0denotethesetofequivalenceclassesinducedbyE.Denotetheequiva-lenceclasstowhichqbelongsby[q].Then8[q]2Q0and8a2A,letq02[q]beastatesuchthatq0;a6=qjunk.Nowdene0[q];a=[q0;a]and0[q];a=q0;a.If8p2[q],p;a=qjunkthen0[q];a=[qjunk]and0[q];a=a.Duetotheassumptions,ifq1;q22[q],andq1;s6=qjunkandq2;s6=qjunk,i.e.,s2domq1domq2A,then[q1;a]=[q2;a]andthustheabovecon-structionproducesadeterministictransducer0=A;Q0;0;0;[q0];Q0withoutanyambiguities.Weshowthat0recognizessequences.Letskandsk+1betwoconsecutivewordsofthesequences.Letgothroughthestatesq0;q1;:::;qnoninputsk,whereqi;ski+16=qjunk.Correspondingly,thetransducer0willgothroughthese-quenceofstates[q0];[q1];:::;[qn]andproducesk+1,sinceforanyi=0:::n)]TJ/F15 11.955 Tf 13.254 0 Td[(1qi;ski+1=qi+1impliesthat0[qi];ski+1=[qi+1]and0[qi];ski+1=qi;ski+1=sk+1i+1.Thusthistransducerrecognizess=s1;s2;:::andthesizeofjQjisequaltothenumberofequivalenceclassesofjEj. LetEbearelationonthesetofstatesofthedeterministictransducerthatsatisesthepremiseoftheProposition3.4.WesaythatEyields0if0isconstructedthroughthealgorithmoutlinedintheProposition3.4usingrelationE. Proposition3.5. LetsbethesequenceofwordsoverAandletsbethetransducerconstructedbythealgorithminProposition2.1thatrecognizess.LetEbeasetthatsatisestheconditionsofProposition3.4,thatwouldproducethesmallestnumberofequivalenceclasses.ThenEyieldsatransducerwithminimalnumberofstatesthatrecognizesthesequences. Proof. Lets=A;Q;;;q0;F,andlet0=A;Q0;0;0;q00;F0denoteaminimaltransducerwiththeminimalnumberofstatesthatwouldrecognizes.Considerfollowingrelation:pqifandonlyifp;q6=qjunkandtherearewordsn1;n22Asuchthatq0;n1=pandq0;n2=qand0q00;n1=0q00;n2.Wesaythatwordsn1andn2correspondtostatespandq. 1. RelationisanequivalencerelationonQnfqjunkg.Itisclearthatitisreexive 18

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andsymmetric.Alsoifq1q2andq2q3then9n01andn02correspondingtoq1andq2andn001andn002correspondingtoq2andq3.Duetothepropertiesofs,itmustbethatn02=n001.Hence0q00;n01=0q00;n02=0q00;n001=0q00;n002.Thusq1q3. 2. ifpqthenifforsomen1;n22Aq0;n1=pandq0;n2=qand0q00;n1=0q00;n2hence0q00;n1a=0q00;n2aandp;aq;a8a2dompdomqA. 3. Ifpqthentherearen1andn2asdescribedabove.Letw2dompdomqhencep;w6=qjunkandq;w6=qjunk.Fromthedenitionofsitfollowsthat9n01;n022Asuchthatn1wn012sandn2wn022s.Itfollowsthat00q00;n1;w=00q00;n2;w.Sinceoutputsofbothtransducersmustagreeonn1wn01andn2wn02itfollowsthatp;w=q;w.Thuspq.Sinceforeachp2Qnfqjunkg,p2Aandq0;p=p,andthewordpwiththispropertyisuniqueitfollowsthattheequivalenceclassesofcanbeputinonetoonecorrespondencewiththesubsetofstatesof0asfollows:Leteach[q]correspondto0q00;w,wherewissuchthatq0;w=q.NowconstructanequivalencerelationEbyaddingqjunkintoanyoneoftheequivalenceclassesoftherelation.ItfollowsthatEsatisesthepremiseoftheProposition3.4.ThusEyieldstransducerwithminimalnumberofstatesthatrecog-nizess. Thusitwasshownthat,ifforagivensequences,sisatransducerthatrecog-nizespreciselyswithpropertiesgiveninproposition2.1thenitispossibletoconstructaminimaltransducerthroughrelationsimilarlyasitwasdoneintherstsection.Themaindierenceisthatinthiscasethingsarealittlemorecomplicatedsinceisnotanequivalencerelation. 19

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4Localpicturelanguages.4.1IntroductionLetLdenotetheinputlanguageofthetransducerandletL2denotethetwo-dimensionallanguageassociatedwithiterativeapplicationsofagiventransducertowordsofL.InthissectionwewillattempttoanalyzetherelationbetweenL2andLfromthepointofviewoflocallanguagesandentropy.Weobservethatfordeterministictransducers,entropyisalwaysequaltozero,howeverthisisnotthecaseifwewillconsiderthetwo-dimensionallanguagecorrespondingtoanondeterministictransducers.4.2Notationanddenitions Denition4.1. ApicturelanguageoveralphabetAisasubsetofAwhereAdenotesthesetofallpossiblerectangularblocksoveralphabetA Denition4.2. AlocalpicturelanguageLoforderkisapicturelanguagesatisfyingB2LifandonlyifFk;kBQk;k,whereQk;kisanitesetofkkblocksandFk;kBdenotesthesetofallkksubblocksofB.Thenotationusedinthissectionisillustratedusingthefollowinggure: k l i j 20

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Figure4:TransducersuchthatLisalocal,butL2isnot q3 a a b a b a q0 q1 Figure5:Transducergeneratingalocalpicturelanguage local?IfL2islocalwithQk;kthentherstthingthatcomestomindistoconsideralocallanguagewithsetofallowedwordsequaltothesetofallwordsinLoflengthk.HoweveritmayhappenthatthereareblocksB1andB2inQk;kwithrstrowsw1andw2respectively,suchthatw1andw2overlap:w1[][2:::k]=w2[][1:::k)]TJ/F15 11.955 Tf 12.041 0 Td[(1],butB1andB2donot,i.e.,B1[][2:::k]6=B2[][1:::k)]TJ/F15 11.955 Tf 11.955 0 Td[(1].Example:LetB1=abaaandB2=abbaandbethetransducerdepictedonFigure5.NotethatL2islocalalmostlocal.Infact,L2nfB2AjF2;2B=fgg=fB1;B2g.Also,B1andB2donotoverlap,butw1=ba,w2=aado.Astheaboveexampleshows,weneedtoenlargethesetofallowedwordsby 23

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includingwordsinLoflengthk+1.Thisway,ifthetwowordsoflengthkoverlapasdidw1andw2above,theirresultingword,i.ew=w1twheret=w2[][k]willbepresentinthelocallanguageonlyifw2L,which,sinceL2islocalpicturelanguage,makessurethatcorrespondingB1andB2alsooverlap.Moreformallywehavethefollowingproposition: Proposition4.1. LetM=fw2Ljkwisundenedg.IfL2isalocalpicturelanguageoforderk,thenLnMmustbealocallanguagewiththesetofallowedwordsP=fwjw2LnMandjwj=korjwj=k+1g. Proof. LetHAbealocallanguagewithPasthesetofallowedwords.Ifw2Hwherejwj=k,thenw2LnMbydenitionofH.Ifjwj>kthenconsiderthefollowingconstruction:B=kw[1:::k]kw[2:::k+1][][1]:::kw[jwj)]TJ/F21 11.955 Tf 17.933 0 Td[(k:::jwj][][1]Sincew[1:::k+1]2LnM,duetotheassumptionskw[1:::k+1]isdened.SinceL2islocalpicturelanguage,Fk;kkw[1:::k+1]=fkw[1:::k];kw[2:::k+1]g:ThusB=kw[1:::k+1]:::kw[jwj)]TJ/F21 11.955 Tf 17.849 0 Td[(k:::jwj][][1].ContinuinginthiswaygetthatB=kw.Ifw2LnM,thenkwisdened.Henceifuisanyfactorofw,i.e.Fk;kkw[jvj:::jvuj]Fk;kkwQk;k:Thusifjuj=kthenu2LnMandifjuj=k+1,u2LnM.Hencew2H. Corollary4.1. IfL2isalocalpicturelanguageandLLthenLislocal. Proof. Sincekwisdenedforeveryw,M=fw2Ljkwisundenedg=fg. 24

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q0 q3 q1 q2 1 1 3 3 2 3 1 2 1 0 0 1 0 0 0 0 0 0 Figure6:Transducergeneratingnon-locallanguage andsupposethatthereisatransducersuchthatL2equalsthesetofallpossibletwodimensionalblocksobtainedthroughtranslationsofagivensetoftiles.Asblocks101111111111and101111101111showdenotetheseblocksasC2andC3respectively,thisisnotpossiblesinceifsuchtransducerwouldexist,thenmustacceptC3[][2]andoutputword111111,asitproducesoneofthevalidtilings.Howeverthisisimpossible,sinceC3[][2]=C2[][2],andthusoutputs111111asathirdrowinC2,whichisnotoneofthevalidtilings.4.5EntropyofL2 Denition4.5. ForLAtheentropyofLishL=limn!1sup1 n2logjBn;nj;whereBn;n=fC2LjCisannnblockg. 26

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ItisclearthatentropyofL2is0foranydeterministictransducer,sinceforeachnitcontainsatmostjAjnblockstoeachw2L2correspondsuniqueblockofheightn.However,ifL2isatwodimensionallanguagecorrespondingtoanondeterministictransducer,entropymaynolongerbe0.Forexample,lettransducer=A;fqg;;fqg;fqgwith=fq;a;b;qja;b2Ag,i.e.consistsofonestatewithallpossibletransitions.HenceL2=Aandforeachn,jBn;nj=jAjn2.ThushL2=hA=logjAj.Foragiventransducerandw2Ldenedegw=jwj,thenumberofdistinctwordsthatcanbeoutputtedbyoninputw,andextendthisnotationtosetsbydegS=sups2Sfdegsg.ThenthenumberofdistinctnnblockscontainedinL2withwjwj=nasarstrowisboundedabovebythefollowingexpression:nwn)]TJ/F19 7.97 Tf 6.586 0 Td[(2Yi=0degiw:Ifforallw2Lwithjwj=nhavethatdegwnkthenhL2=limn!1sup1 n2logBn;nlimn!1sup1 n2logXjwj=nn)]TJ/F19 7.97 Tf 6.586 0 Td[(2Yi=0degiwlimn!1sup1 nlogjAjnk=0:Thusifthereexistsksuchthatforanyw2L,wjwjk,i.e.jwjhasapolynomialbound,theentropyisstillequaltozero. Proposition4.2. ForanyregularlanguageLthereissuchthatL=LandhL2=hL Proof. LetMdenoteaFSA,suchthatLM=L,andletbeitstransitionre-lation.ConstructtransducerfromMbyredeningtransitionrelationas0=fq1;a;s;q2jq1;a;q22;s2Ag.ThusL=L. 27

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NowconsiderL2.NotethatforB2L2,thelast,kthrowmaynotbeinL.LetBk;kdenotethesetofalltwodimensionalblocksinL2ofsizekkandB0p;kdenotethesetofallpkblocksB2L2withB[p][]2LB[p][]denotespthrowofblockB.ThusitmustbetruethatjBk;kjjB0k)]TJ/F19 7.97 Tf 6.586 0 Td[(1;kjjAjkjB0k;kjjAjk.Hence,hL2=limsupk!11 k2logjBk;kj.1limsupk!11 k2logjB0k;kjjAjk4.2=limsupk!11 k2logjB0k;kj.3Now,sincejB0k;kjjBk;kj,havethathL2=limsupk!11 k2logjB0k;kjAlso,jB0k;kj=jLAkjksinceB0k;k=fB2AjB[i][]2LAk;i=1:::kg.ThushL2=limsupk!11 klogjLAkj=hL Corollary4.2. LetLbearegularlanguage.ThentheentropyofLisanupperboundfortheentropyofL2foranytransducerwithL=L. Proof. LetL2Regandlet=A;Q;;q0;FbesuchthatL=L.Thenconsider0=A;Q;0;q0;F,with0=fq;a;s;pjs2Aandq;a;t;p2;forsomet2Ag.Hence0andhL2hL20.Bytheproofoftheproposition4.2,hL20=hLandclaimfollows. 28

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5ConclusionInthisworkwehaveattemptedtoanalyzesequencesthatcanbeobtainedbyiteratingatransduceronwordsofitsinputlanguage.Wecalledsuchsequencestrans-ducerrecognizable.Forthesequencesthatarenottransducerrecognizablewehavedevelopedanotionofrecognitionincontextandshownthatallniteandperiodicsequencesarerecognizedincontext.Asnotedintheintroduction,transducerrecog-nizablesequencescanberelatedtotheblocksconstructedofWangtiles.EachoftheWangtilescorrespondstoaspecictransitioninthetransducer.Whenthetransducerisdeterministic,eachtileisuniquelydeterminedbyitsleftandbottomedge.ThisisabenecialpropertywhenprocessofassemblyofWangtilesinblocksissimulatedbyDNAstrands,asithelpstoreducethenumberofincorrectpartialtilings[13].ToeachnondeterministictransducerweassociatedatwodimensionallanguageL2.InthecasewhenL2isalocalpicturelanguage,weinvestigatedpropertiesofLthatthisconditioninduces.WehaveshownarelationbetweentheentropyofL2andL.ThisrelationsuggestsawiderangeofthepossiblevaluesfortheentropyofL2.Characterizationofpatternsthatcanbegeneratedbyiterationoftransducersmaybeofinterestinapplications,inparticular,inalgorithmicself-assemblyoftwo-dimensionalarrayswithDNAtiles.Also,itmaybeofinteresttoinvestigatesomedecidabilityquestionsfortheselanguagesaswell:forexample,givenatransducergeneratedlanguage,isthereanmn-blockinthelanguageforeverym;n?Therela-tionshipoftheclassoftransducergeneratedlanguageswiththeclassofunambiguousornon-deterministicpicturelanguagesaswellastransitivityandmixingpropertiesoftheselanguagesremaintobeinvestigatedaswell. 29

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23,921-937.[11]Rozenberg,SalomaaEds.HandbookofFormalLanguages:Volume3.Be-yondWords.2002.[12]H.Wang,Notesonclasstilingproblems.FundamentalMathematics,82,295-305.[13]E.Winfree,AlgorithmicSelf-AssemblyofDNA:TheoreticalMotivationsand2DAssemblyExperiments.JournalofBiomolecularStructureandDynamics,11S2,263-270.[14]E.Winfree,F.Liu,L.A.Wenzler,N.C.Seeman,Designandself-assemblyoftwo-dimensionalDNAcrystals,Nature,394,539-544. 31

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